Superposition with subunitary powers in Sobolev spaces

Petru Mironescu (1)

1 EDPA - Équations aux dérivées partielles, analyse

Let $0$<$a$<$1$ and set $\Phi (t)=|t|^a$, $t\in {\mathbb R}$. We prove that the superposition operator $u\mapsto \Phi (u)$ maps the Sobolev space $W^{1,p}({\mathbb R}^n)$ into the fractional Sobolev space $W^{a,p/a}({\mathbb R}^n)$. We also investigate the case of more general nonlinearities.

Mots clés: Sobolev spaces; functional calculus

Domaines: Analyse classique [math.CA]

Petru Mironescu. Superposition with subunitary powers in Sobolev spaces. Comptes Rendus. Mathématique, 2015, 353 (6), pp.483--487. ⟨hal-01069611v3⟩